Tomus 55 (2019), 83–96
TOPOLOGICAL DEGREE THEORY IN FUZZY METRIC SPACES
M.H.M. Rashid
Abstract. The aim of this paper is to modify the theory to fuzzy metric spaces, a natural extension of probabilistic ones. More precisely, the modifica- tion concerns fuzzily normed linear spaces, and, after defining a fuzzy concept of completeness, fuzzy Banach spaces. After discussing some properties of mappings with compact images, we define the (Leray-Schauder) degree by a sort of colimit extension of (already assumed) finite dimensional ones. Then, several properties of thus defined concept are proved. As an application, a fixed point theorem in the given context is presented.
1. Introduction and preliminaries
Topological degree theory is a generalization of the winding number of a curve in the complex plane. It can be used to estimate the number of solutions of an equation, and is closely connected to fixed-point theory. When one solution of an equation is easily found, degree theory can often be used to prove existence of a second, nontrivial, solution. There are different types of degree for different types of maps: e.g. for maps between Banach spaces there is the Brouwer degree inRn, theLeray-Schauder degreefor compact mappings in normed spaces, the coincidence degree and various other types. There is also a degree for continuous maps between manifolds. Topological degree theory has applications in complementarity problems, differential equations, differential inclusions and dynamical systems [10].
Many problems in science lead to the equationx=y in infinite dimensional spaces rather than to the finite dimensional case. In particular, ordinary and partial differential equations, and integral equations can be formulated as abstract equations on infinite dimensional spaces of functions. In 1934, Leray and Schauder [18] generalized Brouwer degree theory to a finite Banach space and established the so-called the Leray Schauder degree. It turns out that the Leray Schauder degree is a very powerful tool in proving various existence results for nonlinear partial differential equations (see [15], [18], [19], [21], etc.).
The Leray Schauder degree theory is very useful in solving an operator equation of the type (I−S)x = y, where S is compact. In many applications S is not
2010Mathematics Subject Classification: primary 54H25; secondary 47H05, 47H09, 47H10.
Key words and phrases: fuzzy metric space,t-norm ofh-type, topological degree theory.
Received October 16, 2017, revised August 2018. Editor A. Pultr.
DOI: 10.5817/AM2019-2-83
compact, so one may ask it is possible to give an analogue of the Leray Schauder theory in the noncompact case. In 1936, Leray [17] constructed an example to show that it is impossible to define a degree theory for mappings with only a continuity condition.
To solve an infinite dimensional equationSx=y, a very natural method is to approximate the original equation by finite dimensional equations, as we have seen in the Leray Schauder theory. The well-known Galerkin method has proved to be a very efficient tool in finite dimensional approximation. In the 1960s, Browder and Petryshyn systematically studied the finite dimensional method for a large class of mappings, which they called A-proper mappings, and they developed a similar theory to the Brouwer degree.
The question of stability in optimization deals with what happens to an opti- mization problem when the elements of the problem are in some way deformed.
As being expressed by Felix E. Browder, the concept of degree of a mapping, in all its different forms, is one of the most effective tools for studying the properties of the existence and multiplicity of solutions of nonlinear equations. Historically, the well known topological degree is a useful tool in applied mathematics, for example to prove that some nonlinear equations have solutions and to investigate the stability by using the continuation method. The notion of the degree was first introduced explicitly by Brouwer in 1912 in the case of finite dimensional spaces.
Leray and Schauder extended this theme in 1934 to the context of Banach spaces and mappings of the form f =I−g, withIthe identity andga compact mapping (we refer to [6], [12] and [18] for a wide bibliography on the subject.) Afterwards many authors defined and developed the topological degree theory for various classes of non-compact nonlinear mappings between Banach spaces. For references on these notions see [1, 2, 3], [5, 6, 7, 8, 9, 11], [13, 14, 16] and [12].
In recent years, many great developments has been made in the theory and applications of fuzzy metric spaces. In 1960, B. Schweizer and A. Sklar [23] gave a description of the topological structure for a special class of probabilistic metric spaces. In 1983, B. Schweizer and A. Sklar [24] summarized and presented the generally developing situation in this field up-to-date. In H. Sherwood [25] has pointed out the ordinary probability spaceis a special case ofprobabilistic metric spaceand as known that theprobabilistic metric spaceis a special case of thefuzzy metric space[22]. This implies that the study of theory and applications relevant to fuzzy metric space has important practical significant.
As is known to the researchers in this subject, theLeray-Schauder topological degree theory is a forceful tool in the research of operator theory in normed spaces.
This motivates us to establish and study the Leray-Schauder topological degree in fuzzy metric spaces.
Definition 1.1 ([24]). A binary operationT: [0,1]×[0,1]→[0,1] is said to be a continuous t-norm if ([0,1], T) is a topological monoid with unit 1 such that T(a, b)≤T(c, d) whenever a≤c,b≤dfor alla,b,c,d∈[0,1].
Some typical examples of t-norm are the following:
T(a, b) =ab , (product) T(a, b) = min{a, b}, (minimum) T(a, b) = max{a+b−1,0}, (Lukasiewicz) T(a, b) = ab
a+b−ab, (Hamacher)
Definition 1.2. [4] Let X be a linear space over K (field of real or complex numbers). A fuzzy subset N ofX×R(R, the set of real numbers) is called a fuzzy norm onX if and only if for allx, u∈X andc∈K.
(FN1) For allt∈Rwitht≤0,N(x, t) = 0,
(FN2) for allt∈Rwitht >0,N(x, t) = 1,if and only ifx= 0, (FN3) for allt∈Rwitht >0,N(cx, t) =N
x,|c|t
, ifc6= 0,
(FN4) for alls,t∈R, x,u∈X,N(x+u, s+t)≥T{N(x, s), N(u, t)}, (FN5) N(x,·) is a non-decreasing function ofRand lim
t→∞N(x, t) = 1.
The pair (X, N) will be referred to as a fuzzy normed linear space (breifly FNLS).
Theorem 1.3 ([20]). Let(X, N, T)be a FNLS. For x∈X,r∈(0,1),t >0, we define the open ball
Bx(r, t) :={y∈X:N(x−y, t)> r}. Then
τA:={A⊂X:x∈A⇐⇒ ∃t >0, r∈(0,1) :Bx(r, t)⊂A}
is a topology on X. Moreover, if the t-norm T satisfies sup
t∈(0,1)
T(t, t) = 1, then (X, τN)is Hausdorff.
Theorem 1.4 ([20]). Let (X, N, T) be a FNLS. Then (X, τN) is a metrizable topological vector space.
Definition 1.5 ([20]). Let (X, N, T) be a FNLS and{xn}be the sequence in X.
(1) The sequence{xn} is said to be convergent if there existsx∈X such that
t→∞lim N(xn−x, t) = 1, for allt >0.
In this case x is called the limit of the sequence {xn} and we denote
n→∞lim xn=xorxn→x.
(2) The sequence{xn} is called Cauchy sequence if
n→∞lim N(xn+p−xn, t) = 1 for allt >0 and allp∈N.
(3) (X, N, T) is said to be complete if every Cauchy sequence inXis convergent to a point inX. A complete FNLS will be called a fuzzy Banach space.
Definition 1.6 ([24]). Let (X, N, T) be a fuzzy normed linear space.
(a) A sequence{xn} inX is τ-convergent to x∈X if for any >0, λ >0, there exists a positive integerk=k(, λ) such that
N(xn−x, )>1−λ whenevern≥k. In this case, we writexn
−→τ x.
(b) A sequence{xn}inX is aτ-Cauchy sequence if for any >0,λ >0, there exist a positive integerk=k(, λ) such that
N(xn−xm, )>1−λ whenevern, m≥k.
(c) (X, N, T) is said to be τ-complete if every τ-Cauchy sequence in X is τ-convergent to some point inX.
2. Results
Definition 2.1. Let (X, N, T) be a a fuzzy normed space andDbe a subset ofX. A mappingA:D→X is said to be compact ifA(D) is a compact subset ofX.
Lemma 2.2. Let (X, N, T) be a fuzzy normed space, T is a t-norm satisfying T(t, t)≥t for allt∈[0,1], Ωbe a nonempty subset ofX,S: Ω→X be a compact continuous mapping. Then for any neighborhood of θ,u(, λ), >0,λ >0, there exists a finite dimension-valued compact mapping S,λ such that
Sx−S,λ∈u(, λ), x∈Ω.
Proof. Since S: Ω → X is compact, S(Ω) is compact subset of X. For any neighborhood ofθ,u(, λ), >0,λ >0, there existy1, y2, . . . , ym∈S(Ω) such that S(Ω)⊂ ∪mi=1(yi+u(, λ)). Letting
λi(x) = max{0, − {t, N(Sx−yi, t)>1−λ}}, x∈Ω, i= 1, . . . , m , we prove that for eachx∈Ω, there exists somei0such thatλi0(x)>0, 1≤i0≤m.
In fact, sinceSx∈S(Ω)⊂ ∪mi=1(yi+u(, λ)), there exists ani0, 1≤i0≤m, such that Sx∈yi0+u(, λ), i.e., N(Sx−yi0, t)>1−λ. By the left continuity of N there exist i0 < such thatN(Sx−yi0, t) >1−λ. Hence we have λi0(x) >0.
Denote
φ(x) =
m
X
i=1
λi(x).
Then for anyx∈Ω, we haveφ(x)6= 0. Now we define a mappingS,λ: Ω→X as follows:
S,λ(x) =
m
X
i=1
λi(x) φ(x)yi.
Now, we prove that S,λsatisfies the requirements of the lemma. For this purpose, it suffices to prove that λi,i= 1,· · · , m, is a continuous function, i.e., we show that
pi(x) = inf{t:N(Sx−y, t)>1−λ}
is a continuous function. Ifxn−→τ x, it is easy to see that
pi(xn)≤pi(x0) + inf{t:N(Sxn−Sx0, t)>1−λ}, pi(x0)≤pi(xn) + inf{t:N(Sx0−Sxn, t)>1−λ}.
Hence we have
|pi(xn)−pi(x0)| ≤inf{t:N(Sx0−Sxn, t)>1−λ}.
If the right side of the preceding expression were not convergent to 0 asn→ ∞, then there would exist an0>0 such that given positive integerN, there exists ann0> N such that
inf{t:N(Sxn0−Sx0, t)>1−λ}> 0
and consequently, we have
(2.1) N(Sxn0−Sx0, 0)≤1−λ . SinceS is continuous,Sxn
−→τ Sx0, and so we have
n→∞lim N(Sxn0−Sx0, 0) = 1,
which contradicts (2.1). Thus, it follows that it getspi(xn)→pi(x0) asn→ ∞, i= 1, . . . , m. By (FN4), we have
N(Sx−S,λx, )≥ min
1≤i≤m,λi(x)6=0{N(Sx−yi, )}>1−λ .
This implies thatSx−S,λx∈u(, λ) for allx∈Ω. Moreover, obviously,S,λ is
compact. This achieves the proof.
Lemma 2.3. Let (X, N, T)satisfy all the conditions of Lemma 2.2. LetΩ be a nonempty open subset of X and S: Ω → X be a compact continuous mapping.
ThenR=I−S is a closed mapping.
Proof. The conclusion can be proved immediately. The details are omitted here.
Definition 2.4. Let (X, N, T) be a fuzzy normed space,T is a t-norm satisfying T(t, t)≥tfor allt∈[0,1]. Let Ω be a nonempty open subset ofXandS: Ω→X be a compact continuous mapping. LetR=I−S andp∈X\R(∂Ω). By Lemma 2.3, R is a closed mapping, R(∂Ω) is a closed subset of X, and, consequently, there exists a neighborhood ofθ,u(, λ), such that
(p+u(, λ))∩R(∂Ω) =∅.
By Lemma 2.2, there exists a finite dimension subspaceX(n)ofX withp∈X(n) and a continuous compact mappingSn: Ω→X(n)such thatN(Sx−Snx, )>1−λ for all x∈ Ω. Letting Ωn = Ω∩X(n) andRn =I−Sn, we are going to prove p /∈Rn(∂Ω).
In fact, if there exists somex0∈∂Ω such thatp=Rnx0, then we have N(Rx0−p, ) =N(Sx0−Rnx0, ) =N(Sx0−Snx0, )>1−λ .
This contradicts (p+u(, λ))∩R(∂Ω) =∅. Beside, since (I−(I−Sn))(Ωn) is a compact set, the topological degree degn(Rn,Ωn, p) in finite dimensional space X(n)is significant. We define theLeray-Schauder topological degree ofR as follows:
(2.2) Deg (R,Ω, p) = degn(Rn,Ωn, p).
In order to explain the topological degree defined by (2.2) is significant, it suffices to show that it is independent of the choice of the neighborhood of θ,u(, λ), the spaceX(n)and the mappingSn.
First, we prove that, whenu(, λ) is given, Deg (R,Ω, p) is independent of the choice of X(n) andSn. In fact, if X(m) andRm also satisfy the requirements in Definition 2.4, now we prove the following expression holds:
(2.3) degn(Rn,Ωn, p) = degm(Rm,Ωm, p).
LettingX(l) be the linear sum ofX(n) andX(m), Ωl=X(l)∩Ω and noting that Sn can be seen as a mapping from Ω→X(l), we know thatRn is a mapping from Ωl intoX(l). By the reduced theorem of topological degree, we have
degl(Rn,Ωl, p) = degn(Rn,Ωn, p). Similarly, we can prove that
degl(Rn,Ωl, p) = degm(Rm,Ωm, p). Next, we prove that
degl(Rn,Ωl, p) = degl(Rm,Ωl, p). Write
ht(x) =tRn(x) + (1−t)Sm(x).
If there exists at0∈[0,1],x0∈∂Ω such thatp=ht0(x0), then we have N(Rx0−p, ) =N(Rx0−t0Rn(x0)−(1−t0)Rm(x0), )
=N(t0Snx0+ (1−t0)Smx0−Sx0, )
≥T(N(t0(Snx0−Sx0), t0), N((1−t0)(Smx0−Sx0),(1−t0)))
>1−λ ,
which is a contradiction. This implies thatp /∈ ht(∂Ω) for all t ∈ [0,1]. By the homotopy inveriance of topological degree in finite dimensional spaces, we have
degl(Rn,Ωl, p) = degl(Rm,Ωl, p).
This shows that (2.3) is true.
Next, we prove that Deg (R,Ω, p) is independent of the choice ofu(, λ). Suppose that there exists neighborhood of θ, u1(1, λ1), satisfying all the conditions of Definition 2.4. Taking
0< 0≤min{, 1}, 0< λ0≤min{λ, λ1},
it follows that u(0, λ0) also satisfies all the conditions of Definition 2.4 foru(, λ), u(1, λ1) andu(0, λ0), respectively, by the choice of0,λ0, it is obvious thatRl,
Ωlsatisfy all the conditions of Definition 2.4 for both u(, λ) andu1(1, λ1), too. It follows from (2.3) that
degn(Rn,Ωn, p) = degl(Rl,Ωl, p), degm(Rm,Ωm, p) = degl(Rl,Ωl, p). Hence we have
degm(Rm,Ωm, p) = degn(Rn,Ωn, p).
Thus, summing up the above explanation, we know that the topological degree defined by (2.2) is significant.
In the sequel of this section, we study the properties of topological degree defined by (2.2).
Theorem 2.5. The topological degree defined by (2.2)has the following properties:
(a) Deg (I,Ω, p) = 1for all p∈Ω,
(b) If Deg (R,Ω, p)6= 0, then the equation R(x) =phas a solution in Ω, (c) If H(t, x)is a continuous compact mapping defined on[0,1]×Ωandp /∈
(I−H(t,·))(∂Ω)for allt∈[0,1], thenDeg (I−H(t,·),Ω, p)is independent of t∈[0,1],
(d) IfΩ1,Ω2 are two disjoint open subsets ofΩandp /∈R(Ω\(Ω1∪Ω2)), then Deg (R,Ω, p) = Deg (R,Ω1, p) + Deg (R,Ω2, p),
(e) If Ω0 is an open subset ofΩandp /∈R(Ω\Ω0), then Deg (R,Ω, p) = Deg (R,Ω0, p), (f) If p /∈R(∂Ω), then
Deg (R,Ω, p) = Deg (R−p,Ω, θ).
Proof. (a) and (f) can be obtained from Definition 2.4 immediately.
(b) Suppose that the equationR(x) =phas no solution in Ω. Thenp /∈R(Ω). In view of Lemma 2.3,R(Ω) is a closed subset and hence there exists a neighborhood of θ,u(, λ), such that (p+u(, λ))∩R(Ω) =∅. Take a finite dimension subspaceX(n) of X and a finite dimension-valued continuous compact mappingSn: Ω→X(n) such that
Sx−Snx∈u(, λ), x∈Ω.
LettingRn=I−Sn and Ωn =X(n)∩Ω, by Definition 2.4, we have Deg (R,Ω, p) = degn(Rn,Ωn, p).
If there exist an x0∈Ωn⊂Ω such thatRnx0=p, then we have
N(Rx0−p, ) =N(Rx0−Rnx0, ) =N(Sx0−Snx0, )>1−λ .
This contradicts (p+u(, λ))∩R(Ω) =∅.Thus we havep∈Rn(Ωn), hence we have Deg (S,Ω, p) = deg(Rn,Ωn, p) = 0,
which is a contradiction. This achieves the proof of (b).
(c) First we prove that there exists a neighborhood ofθ, u(, λ), such that the following expression uniformly holds int∈[0,1]:
(p+u(, λ))∩(I−H(t,·))(∂Ω) =∅.
Otherwise, there existn>0,λn>0,n= 1,2, . . . ,withλn→0, n →0 asn→ ∞ andxn∈∂Ω,tn ∈[0,1],n= 1,2, . . ., such that
N(p−xn+H(xn, xn), n)>1−λn.
Since both {tn} and{H(tn, xn)} have convergent subsequences, without loss of generality, we still denote these subsequences by{tn}and{H(tn, xn)}andtn →t0, H(tn, xn)→qasn→ ∞. By (FN5), we have
N(p−xn+q, )≥T N
p−xn+H(tn, xn), 2
, N(q−H(tn, xn), 2)
, it follows that xn →p+q∈∂Ω asn→ ∞. Thus we have
p= (I−H(t0,·))(p+q), which is a contradiction. Therefore the assertion is true.
Besides, by virtue of Lemma 2.2, there exist a finite dimension subspaceX(n)⊂ X and a finite dimension-valued compact continuous mappingQn: [0,1]×Ω→X(n) such that
H(t, x)−Qn(t, x)∈u(, λ), (t, x)∈[0,1]×Ω. Lettingqt(x) =x−Qn(t, x) and Ωn=X(n)∩Ω, then we have
Deg (I−H(t,·),Ω, p) = degn(qt,Ωn, p), t∈[0,1]. If there existx0∈∂Ωn,t0∈[0,1] such thatqt0(x0) =p, then we have
N(x0−H(t0, x0)−p, ) =N(x0−H(t0, x0)−x0+Qn(t0, x0), )>1−λ , which is a contradiction. Therefore, we know thatp /∈qt(∂Ω) for allt∈[0,1] and hence we have
Deg (I−H(t,·),Ω, p) = degn(qt,Ωn, p) = a constant.
(d) Since Ω\(Ω1∪Ω2) is a closed subset,R(Ω\(Ω1∪Ω2)) is also a closed subset.
Hence there exists a neighborhood of θ,u(, λ), such that (p+u(, λ))∩R(Ω\(Ω1∪Ω2)) =∅.
Consequently, we can a finite dimension subspaceX(n)ofXand a finite dimension- -valued continuous compactRn: Ω→X(n) such that for anyx∈Ω,Sx−Snx∈
u(, λ). Letting
Rn=I−Sn, Ωn=X(n)∩Ω1, Ω(1)n =X(n)∩Ω1, Ω(2)n =X(n)∩Ω2, it follows from Definition 2.4 that
Deg (R,Ω, p) = degn(Rn,Ωn, p), Deg (R,Ω1, p) = degn(Rn,Ω(1)n , p), Deg (R,Ω2, p) = degn(Rn,Ω(2)n , p).
It is obvious that Ω(1)n ∩Ω(2)n =∅. Ifp∈Rn(Ωn\(Ω(1)n ∪Ω(2)n )), then there exists anx0such thatRn(x0) =p. However, since we have
N(x0−Sx0−p, ) =N(x0−Sx0−x0+Snx0, )>1−λ , this contradicts
(p+u(, λ))∩R(Ω\(Ω1∪Ω2)) =∅. Hence we havep /∈Rn(Ωn\(Ω(1)n ∪Ω(2)n )) and so
degn(Rn,Ωn, p) = degn(Rn,Ω(1)n , p) + degn(Rn,Ω(2)n , p), that is to say,
Deg (R,Ω, p) = Deg (R,Ω1, p) + Deg (R,Ω2, p).
The conclusion (f) Can be obtained from (d) immediately. This achieves the
proof.
Theorem 2.6. The topological degree defined by (2.2)has the following properties:
(i) If there exist the degrees of R1 and R2 such that p ∈ X \R1(∂Ω) and R1(x) =R2(x)for all x∈∂Ω, then
Deg (R1,Ω, p) = Deg (R2,Ω, p),
(ii) Ifpvaries on every connected component of X\R(∂Ω), then the degree Deg (R,Ω, p) is a constant.
Proof. (i) can be obtained immediately.
(ii) LetV be a connected component ofX\R(∂Ω) andp∈V. Then there exists a neighborhood ofθ, u(0, λ0), such that (p+u(0, λ0))∩R(∂Ω) =∅. Take positive numbers 1, λ1with1< 0,λ1< λ0,q∈V(p+u(1, λ1)), and write
qt(x) =R(x)−t(q−p), 0≤t≤1, x∈Ω.
If there existt0∈[0,1],x0∈∂Ω such thatR(x0)−t0(q−p) =p, then we have N(R(x0)−p, 0) =N(t0(q−p), 0)>1−λ0,
which is a contradiction. Thus it follows thatp /∈qt(∂Ω) for allt∈[0,1]. Therefore we have
Deg (R,Ω, p) = Deg (R−(q−p),Ω, p)
= Deg (S−q,Ω, θ) = Deg (R,Ω, q).
This implies that the mapping Θ :p→Deg (R,Ω, p) is a continuous mapping on V. By a well-known result of general topology, we know that Θ(V) is a connected component. Since Θ is an integer-valued function, for anyp∈V, Deg (R,Ω, p) has
the same value. This achieves the proof.
Theorem 2.7. LetS andS1 be two compact continuous mappings fromΩintoX. If p /∈R1(∂Ω),p /∈R(∂Ω),R1=I−S1,R=I−S and the following condition is satisfied:
(2.4) N(S1x−Sx, t)≥N(x−Sx−p, t), t >0, x∈∂Ω,
then
Deg (R−1,Ω, p) = Deg (R,Ω, p). Proof. Letting
qt(x) =x−Sx−t(S1x−Sx), t∈[0,1], x∈Ω,
we are going to prove p /∈qt(∂Ω) for allt ∈ [0,1]. Suppose that this is not the case. Then there exist some t0 ∈ [0,1] and an x0 ∈ ∂Ω such that qt0(x0) = p.
It follows from the assumptions of theorem that t0 6= 0 and t0 6= 1. In view of x0−Sx0−p=t0(S1x0−Sx0), we have
(2.5) N(x0−Sx0−p, t) =N
S−1x0−Sx0, t t0
, t >0. It follows from (2.5) and the conditions of this theorem that N(S1x0−Sx0, t) =N
S1x0−Sx0, t t0
=· · ·=N
S1x0−Sx0, t tn0
, n= 1,2, . . . . This implies thatN(S1x0−Sx0, t) = 1 for allt >0. By (2.5), we have
N(x0−Sx0−p, t) = 1, t >0,
which shows thatp=x0−Sx0, i.e.,p∈R(∂Ω). This contradictsp /∈R(∂Ω). Thus p /∈qt(∂Ω) for allt∈[0,1] and so we have
Deg (R1,Ω, p) = Deg (R,Ω, p).
This achieves the proof.
Corollary 2.8. Ifθ∈Ω,S1: Ω→X is a continuous compact mapping satisfying the conditions:
x6=S1x , N(S1x, t)≥N(x, t), t >0, x∈∂Ω. Then
Deg (I−S1,Ω, θ) = 1.
Theorem 2.9. LetΩbe an open set withθ∈Ωand letΩbe symmetric with respect to θ. Suppose thatS: Ω→X is a continuous compact mapping andR=I−S. If
S(−x) =−S(x), Sx6=x , x∈∂Ω, then Deg (R,Ω, θ)is an odd number.
Proof. Imitating the proof of Lemma 2.2, for any neighborhood ofθ,u(, λ), >0, λ > 0, we can make a finite dimension-valued continuous compact mappingSn
satisfying the following conditions:
(a) Sn(−x) =−Sn(x) for allx∈∂(Ω∩X(n)), (b) Sx−Snx∈u(, λ) for allx∈Ω.
Since the value of degree deg(Rn,Ω, θ) is odd, the value of degree Deg (R,Ω, θ) is
also odd, where Ωn=X(n)∩Ω.
Now, we shall utilize the theory of topological degree to study some fixed point theorems for mappings in fuzzy normed spaces. Let us assume that thet-normT satisfies the condition T(t, t)≥t for allt∈[0,1].
Theorem 2.10. LetΩbe an open convex subset ofX andS: Ω→X be a compact continuous mapping such thatS(∂Ω)⊂Ω. ThenS has a fixed point inΩ.
Proof. Without loss of generality, we can assume that Sx6= xfor all x∈ ∂Ω (otherwise the conclusion of Theorem has been proved). Takingx0∈Ω and letting H(t, x) =tSx+ (1−t)x0, we know thatH: [0,1]×Ω→X is a continuous compact mapping. Lettingqt(x) =x−H(t, x), we prove that
θ /∈qt(∂Ω), t∈[0,1].
Suppose this is not the case. Then there exist ant1∈[0,1] and anx1∈∂Ω such that qt1(x1) =θ, i.e.,
x1=t1Sx1+ (1−t1)x0.
It is obvious that t16= 0 and t1 6= 1. Since Ω is an open set, there exist0 >0, λ0>0 such thatx0+u(, λ)⊂Ω. BecauseSx1∈Ω, we havex0∈Ω and
(2.6) Sx1−z0∈ 1−t1
t1
u(0, λ0). Next we prove that
(2.7) t1z0+ (1−t1)x0+ (1−t1)u(0, λ0)⊂Ω.
In fact, ifx∈t1z0+(1−t1)x0+(1−t1)u(0, λ0), then there exists somez∈u(0, λ0) such that
x=t1z0+ (1−t1)x0+ (1−t1)z=t1z0+ (1−t1)(x0+z).
Sincex0+u(0, λ0)⊂Ω,x0+z∈Ω. Next sincez0∈Ω and Ω is a convex set, we have x∈Ω. This shows that (2.7) is true. Hence we have
x1=t1Sx1+ (1−t1)x0=t1z0+ (1−t1)x0+t1(Sx1−z0).
It follows from (2.6) thatt1(Sx1−z0)∈(1−t1)u(0, λ0). By (2.7), it follows that x1∈Ω. This contradictsx1∈∂Ω and henceθ /∈qt(∂Ω) for allt∈[0,1]. Therefore we have
Deg (I−S,Ω, θ) = Deg (I−x0,Ω, θ) = 1,
which implies thatS has a fixed point in Ω. This achieves the proof.
Theorem 2.11. Let Ωbe an open subset ofX withθ∈Ω. IfΩis symmetric with respect to θ and if S:∂Ω→ X is a compact continuous mapping satisfying the following condition:
S(−x) =−Sx, x∈∂Ω. ThenS has a fixed point inΩ.
Proof. The assertion follows from Theorem 2.9 immediately.
Moreover, from Corollary 2.8, we can obtain the following:
Theorem 2.12. LetΩ be an open subset of X withθ ∈Ω. If S:∂Ω→ X is a compact continuous mapping satisfying the following condition:
N(Sx, t)≥N(x, t), x∈∂Ω, t >0. ThenS has a fixed point inΩ.
Theorem 2.13. LetΩ1, Ω2 be two open subsets of an infinite dimension fuzzy normed space (X, N, T), θ ∈ Ω1, Ω1 ⊂ Ω2, where the t-norm T satisfies the condition: T(t, t)≥t for all t∈[0,1]. Suppose that S: Ω2 →X is a continuous compact mapping. If one the following conditions holds:
(i) for any x∈∂Ω1,N(Sx, t)≥N(x, t) for all t≥0, and for any x∈∂Ω2, N(Sx, t)≤N(x, t)for all t≥0,
(ii) for any x∈∂Ω1,N(Sx, t)≤N(x, t) for all t≥0, and for any x∈∂Ω2, N(Sx, t)≥N(x, t)for all t≥0.
ThenS has at least a fixed point inΩ2\Ω1.
In order to give the proof of Theorem 2.13, we need the following lemma:
Lemma 2.14. Let Ω be an open subset of an infinite dimension fuzzy normed space (X, N, T) with T(t, t) ≥ t for all t ∈ [0,1]. Suppose that S: Ω → X is a continuous compact mapping satisfying the following conditions:
(i) θ /∈S(∂Ω),
(ii) Sx6=µxfor allµ∈[0,1]andx∈∂Ω.
ThenDeg (I−S,Ω, θ) = 0.
Proof. First we prove thatθ /∈ ∪µ∈[0,1](µI−S)(∂Ω). Suppose this is not the case.
Then there existxn ∈∂Ω,µn ∈[0,1] such thatµnxn−Sxn→θ asn→ ∞. Since S is a compact continuous mapping, there exist subsequences{µnk} ⊂ {µn}and {xnk} ⊂ {xn}such thatµnk →µ0∈[0,1],Sxnk→y0∈X.
(a) Ifµ0= 0, thenSxnk →θ, which contradicts condition (i).
(b) Ifµ06= 0, thenxnk →y0/µ0∈∂Ω and hence we have Sy0
µ0
=y0=µ0· y0
µ0
.
This contradicts the condition (ii) and so the the assertion holds.
Therefore there exists some neighborhood ofθ,u(, λ), >0,λ >0, such that (2.8) u(, λ)∩ ∪µ∈[0,1](µI−S)(∂Ω) =∅.
By Lemma 2.2 and Definition 2.4, there exists a finite dimension-valued compact continuous mapping Sn: Ω→X(n) such that
Sx−Snx∈u(, λ), x∈∂Ω, Deg (I−S,Ω, θ) = deg(I−Sn,Ωn, θ),
where Ωn= Ω∩X(n). By assumption, (X, N, T) is infinitely dimensional and hence there exists an e16=θ ande1 ∈/ X(n). Letting X(n+1) =span{e1, X(n)}, we can assume that Sn is a mapping from Ω intoX(n+1). Put Ωn+1 = Ω∩X(n+1). By Definition 2.4, it follows that
(2.9) Deg (I−S,Ω, θ) = deg(I−Sn,Ωn+1, θ).
Next, we prove that for any x∈∂Ωn+1 ⊂∂Ω, θ6=µx−Snxfor allµ∈[0,1]. In fact, if there exist some µ0∈[0,1] and anx0∈∂Ωn+1 such thatµ0x0−Snx0=θ,
then we have µ0x0 = Snx0. Since Sx−Snx ∈ u(, λ) for all x ∈ Ω, we have Sx0−µ0x0∈u(, λ). This contradicts (2.8). Thus the assertion is true. Therefore, on Ωn+1, we have
(2.10) deg(I−Sn,Ωn+1, θ) = deg(−Sn,Ωn+1, θ).
However, sinceSn is a mapping from Ωn+1 into X(n), we have deg(−Sn,Ωn+1, θ)
= 0. It follows from (2.9) and (2.10) that
Deg (I−S,Ω, θ) = 0.
This achieves the proof.
Proof of Theorem 2.13. Suppose that the condition (i) is satisfied and S has no fixed point in∂Ω1∪∂Ω2(otherwise, the conclusion of theorem has been shown).
It follows from Corollary 2.8 that
Deg (I−S,Ω1, θ) = 1.
By the assumption, for anyx∈∂Ω2,N(Sx, t)≤N(x, t) for allt ≥0 and hence we have
θ /∈S(∂Ω2) and Sx6=µx, µ∈(0,1].
From Lemma 2.14, it follows that Deg (I−S,Ω2, θ) = 0. Besides, since Deg (I−S,Ω2\Ω1, θ) = Deg (I−S,Ω2, θ)−Deg (I−S,Ω1, θ)
= 0−1 =−1, S has a fixed point in Ω2\Ω1.
If the condition (ii) is satisfied, in the same way, we can prove the assertion
holds too. This achieves the proof.
Acknowledgement. I am grateful to the referee for his valuable comments and helpful suggestions.
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Department of Mathematics and Statistics, Faculty of Science, Mu’tah University, P.O. Box (7), Al-Karak, Jordan E-mail:[email protected]
